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I have this theorem with proof that i need to understand, however the proof in the book i am into is not complete.

Thm: If $(p_1, p_2)$, $(q_1, q_2)$ and $(r_1, r_2)$ are vertices of a triangle, then the area of that triangle is the absolute value of $\frac{1}{2}[(q_1-p_1)(r_2-p_2) - (q_2-p_2)(r_1-p_1)]$.

The proof in the book says that the first shaded triangle should be proven first and it is left for the reader. The first triangle(shaded) should be proven first

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Then the following is given for the proof of the second triangle,which is the goal of the proof: Since $τP,O(P)=(0,0)$, $τP,O(Q)=(a,b)$ and $τP,O(P)=(c,d)$, then the area of $∆PQR$ is also $\frac{1}{2}|ad-bc|$. Substitution then yields the expression given in the statement of the theorem.

However the only idea I have is that the translation that takes $P$ to $Q$ and $P$ to $R$ has equations: $q_1-p_1=a$ , $q_2-p_2=b$ and $r_1-p_1=c$ , $r_2-p_2=d$.

emma
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Ryan Liares Dagasuhan
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1 Answers1

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Those are simply relation between cartesian coordinates, for example with reference to PQ for the triangle on the right we have that

  • distance $PQ$ in $x$ direction $= q_1-p_1$

  • distance $PQ$ in $y$ direction $= q_2-p_2$

Since the shaded triangle is obtained by a translation of the right triangle such that $P\equiv O$, for the shaded triangle we have

  • distance $PQ$ in $x$ direction $a= q_1-p_1$

  • distance $PQ$ in $y$ direction $b = q_2-p_2$

and ths the point $Q$ for the shaded triangle has coordinates $(a,b)$.

user
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